Counting irreducible polynomials with prescribed coefficients over a finite field

TL;DR

This paper develops a combinatorial framework using generating functions to count irreducible polynomials over finite fields with specific coefficients, extending previous results in the area.

Contribution

It introduces a new combinatorial approach with generating functions for counting such polynomials, providing simplified formulas for special cases.

Findings

Extended earlier results on counting irreducible polynomials

Derived simplified expressions for specific cases

Established a general framework using group algebra

Abstract

We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by equivalent classes of polynomials with prescribed coefficients. Simplified expressions are derived for some special cases. Our results extend some earlier results.